A voyage round coalgebras [expository notes]

28 Pages·2016·0.354 MB·English

by  Marco Manetti

A VOYAGE ROUND COALGEBRAS
MARCOMANETTI
Abstract. I found the most ready way of explaining my employment was to ask them how
itwasthattheythemselveswerenotcuriousconcerningearthquakesandvolcanos?-whysome
springswerehotandotherscold?-whythereweremountainsinChile,andnotahillinLaPlata?
These bare questions at once satisfied and silenced the greater number; some, however (like a
fewinEnglandwhoareacenturybehindhand),thoughtthatallsuchinquirieswereuselessand
impious;andthatitwasquitesufficientthatGodhadthusmadethemountains.
Charles Darwin: The voyage of the Beagle.
Notation. WeworkoverafixedfieldKofcharacteristic0.Unlessotherwisespecifiedallthetensor
products are made over K. We denote by G the category of graded vector spaces over K.
The tensor algebra generated by V ∈G is by definition the graded vector space
T(V)= (cid:76) (cid:78)nV
n≥0
endowed with the associative product (v ⊗···⊗v )(v ⊗···⊗v )=v ⊗···⊗v .
1 p p+1 n 1 n
LetV,W ∈G.Thetwistmaptttwww: V⊗W →W⊗V isdefinedbytheruletttwww(v⊗w)=(−1)vww⊗v,
for every pair of homogeneous elements v ∈V, w ∈W.
The following convention is adopted in force: let V,W be graded vector spaces and F: T(V)→
T(W) a linear map. We denote by
Fi: T(V)→(cid:78)iW, F : (cid:78)jV →T(W), Fi: (cid:78)jV →(cid:78)iW
j j
the compositions of F with the inclusion (cid:78)jV →T(V) and/or the projection T(W)→(cid:78)iW.
1. Graded coalgebras
Definition 1.1. A coassociative Z-graded coalgebra is the data of a graded vector space C =
⊕n∈ZCn ∈G and of a coproduct ∆: C →C⊗C such that:
• ∆ is a morphism of graded vector spaces.
• (coassociativity) (∆⊗Id )∆=(Id ⊗∆)∆: C →C⊗C⊗C.
C C
For simplicity of notation, from now on with the term graded coalgebra we intend a Z-graded
coassociative coalgebra.
Definition 1.2. Let (C,∆) and (B,Γ) be graded coalgebras. A morphism of graded coalgebras
f: C →B is a morphism of graded vector spaces that commutes with coproducts, i.e.
Γf =(f ⊗f)∆: C →B⊗B.
The category of graded coalgebras is denoted by GC.
Example 1.3. LetC =K[t]bethepolynomialringinonevariablet(ofdegree0).Thelinearmap
n
(cid:88)
∆: K[t]→K[t]⊗K[t], ∆(tn)= ti⊗tn−i,
i=0
gives a coalgebra structure (exercise: check coassociativity).
Date:September2006.
1
2 MARCOMANETTI
For every sequence f ∈K, n>0, it is associated a morphism of coalgebras f: C →C defined
n
as
n
(cid:88) (cid:88)
f(1)=1, f(tn)= f f ···f ts.
i1 i2 is
s=1 (i1,...,is)∈Ns
i1+···+is=n
The verification that ∆f =(f⊗f)∆ can be done in the following way. Let {xn}⊂C∨ =K[[x]] be
the dual basis of {tn}. Then for every a,b,n∈N we have:
(cid:88)
(cid:104)xa⊗xb,∆f(tn)(cid:105)= f ···f f ···f ,
i1 ia j1 jb
i1+···+ia+j1+···+jb=n
(cid:88) (cid:88) (cid:88)
(cid:104)xa⊗xb,f ⊗f∆(tn)(cid:105)= f ···f f ···f .
i1 ia j1 jb
s i1+···+ia=s j1+···+jb=n−s
Note that the sequence {f }, n≥1, can be recovered from f by the formula f =(cid:104)x,f(tn)(cid:105).
n n
Example 1.4. Let A bea gradedassociative algebrawith product µ: A⊗A→A and C a graded
coassociative coalgebra with coproduct ∆: C → C ⊗C. Then Hom∗(C,A) is a graded associative
algebra by the convolution product
fg =µ(f ⊗g)∆.
We left as an exercise the verification that the product in Hom∗(C,A) is associative. In particular
Hom (C,A)=Hom0(C,A)isanassociativealgebraandC∨ =Hom∗(C,K)isagradedassociative
G
algebra.
Remark 1.5. The above example shows in particular that the dual of a coalgebra is an algebra.
In general the dual of an algebra is not a coalgebra (with some exceptions, see e.g. Example 2.3).
Heuristically, this asymmetry comes from the fact that, for an infinite dimensional vector space V,
thereexistanaturalmapV∨⊗V∨ →(V ⊗V)∨,whiledoesnotexistanynaturalmap(V ⊗V)∨ →
V∨⊗V∨.
Example 1.6. The dual of the coalgebra C =K[t] (Example 1.3) is exactly the algebra of formal
powerseriesA=K[[x]]=C∨.Everycoalgebramorphismf: C →C inducesalocalhomomorphism
of K-algebras ft: A → A. The morphism ft is uniquely determined by the power series ft(x) =
(cid:80) f xn and then every morphism of coalgebras f: C → C is uniquely determined by the
n>0 n
sequence f = (cid:104)ft(x),tn(cid:105) = (cid:104)x,f(tn)(cid:105). The map f (cid:55)→ ft is functorial and then preserves the
n
composition laws.
Definition 1.7. Let (C,∆) be a graded coalgebra; the iterated coproducts ∆n: C → C⊗n+1 are
defined recursively for n≥0 by the formulas
∆0 =Id , ∆n: C −∆→C⊗C −I−dC−−⊗−∆−n−−→1 C⊗C⊗n =C⊗n+1.
C
Lemma 1.8. Let (C,∆) be a graded coalgebra. Then:
(1) For every 0≤a≤n−1 we have
∆n =(∆a⊗∆n−1−a)∆: C →(cid:78)n+1C.
(2) For every s≥1 and every a ,...,a ≥0 we have
0 s
(∆a0 ⊗∆a1 ⊗···⊗∆as)∆s =∆s+Pai.
(3) If f: (C,∆)→(B,Γ) is a morphism of graded coalgebras then, for every n≥1 we have
Γnf =(⊗n+1f)∆n: C →(cid:78)n+1B.
Proof. [1] If a=0 or n=1 there is nothing to prove, thus we can assume a>0 and use induction
on n. we have:
(∆a⊗∆n−1−a)∆=((Id ⊗∆a−1)∆⊗∆n−1−a)∆
C
=(Id ⊗∆a−1⊗∆n−1−a)(∆⊗Id )∆
C C
=(Id ⊗∆a−1⊗∆n−1−a)(Id ⊗∆)∆
C C
=(Id ⊗(∆a−1⊗∆n−1−a)∆)∆=∆n.
C
A VOYAGE ROUND COALGEBRAS 3
[2] Induction on s, being the case s=1 proved in item 1. If s≥2 we can write
(∆a0 ⊗∆a1 ⊗···⊗∆as)∆s =(∆a0 ⊗∆a1 ⊗···⊗∆as)(Id ⊗∆s−1)∆=
C
(∆a0 ⊗(∆a1 ⊗···⊗∆as)∆s−1)∆=(∆a0 ⊗∆s−1+Pi>0ai)∆=∆s+Pai.
[3] By induction on n,
Γnf =(Id ⊗Γn−1)Γf =(f ⊗Γn−1f)∆=(f ⊗(⊗nf)∆n−1)∆=(⊗n+1f)∆n.
B
(cid:3)
Definition 1.9. Let (C,∆) be a graded coalgebra and p: C → V a morphism of graded vector
spaces. We shall say that p is a system of cogenerators of C if for every c ∈ C there exists n ≥ 0
such that (⊗n+1p)∆n(c)(cid:54)=0 in (cid:78)n+1V.
Example 1.10. In the notation of Example 1.3, the natural projection K[t]→K⊕Kt is a system
of cogenerators.
Proposition 1.11. Let p: B →V be a system of cogenerators of a graded coalgebra (B,Γ). Then
every morphism of graded coalgebras φ: (C,∆)→(B,Γ) is uniquely determined by its composition
pφ: C →V.
Proof. Let φ,ψ: (C,∆) → (B,Γ) be two morphisms of graded coalgebras such that pφ = pψ. In
order to prove that φ=ψ it is sufficient to show that for every c∈C and every n≥0 we have
(⊗n+1p)Γn(φ(c))=(⊗n+1p)Γn(ψ(c)).
By Lemma 1.8 we have Γnφ=(⊗n+1φ)∆n and Γnψ =(⊗n+1ψ)∆n. Therefore
(⊗n+1p)Γnφ=(⊗n+1p)(⊗n+1φ)∆n =(⊗n+1pφ)∆n =
=(⊗n+1pψ)∆n =(⊗n+1p)(⊗n+1ψ)∆n =(⊗n+1p)Γnψ.
(cid:3)
Definition 1.12. Let (C,∆) be a graded coalgebra. A linear map d ∈ Homn(C,C) is called a
coderivation of degree n if it satisfies the coLeibniz rule
∆d=(d⊗Id +Id ⊗d)∆.
C C
A coderivation d is called a codifferential if d2 =d◦d=0.
Example 1.13. For every k ≥−1 consider the differential operator
(cid:18)d (cid:19)k+1
f : K[t]→K[t], f =t .
k k dt
Then every f is a coderivation with respect the (associative, cocommutative) coproduct
k
n (cid:18) (cid:19)
∆˜: K[t]→K[t]⊗K[t], ∆˜(tn)=(cid:88) n ti⊗tn−i.
i
i=0
Using the definition of binomial coefficients
(cid:18) (cid:19) k−1 (cid:18) (cid:19)
n 1 (cid:89) n
= (n−i), =1,
k k! 0
i=0
we have for every n≥0 and every k ≥0
∆˜(fk−1(tn)) =∆˜((cid:18)n(cid:19)tn−k+1)=(cid:88)(cid:18)n(cid:19)(cid:18)n−k+1(cid:19)ti⊗tn−k−i+1,
k! k k i
i≥0
(fk−1⊗Id)∆˜(tn) =(cid:88)(cid:18)n(cid:19)(cid:18)j(cid:19)tj−k+1⊗tn−j =(cid:88)(cid:18) n (cid:19)(cid:18)i+k−1(cid:19)ti⊗tn−k−i+1,
k! j k i+k−1 k
j≥k i≥0
(Id⊗fk−1)∆˜(tn) =(cid:88)(cid:18)n(cid:19)(cid:18)n−i(cid:19)ti⊗tn−k−i+1,
k! i k
i≥0
4 MARCOMANETTI
and the conclusion follows from the straightforward equality
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
n n−k+1 n i+k−1 n n−i
= + .
k i i+k−1 k i k
Notice that
[f ,f ]=f ◦f −f ◦f =(n−m)f .
n m n m m n n+m
Notice that the Lie subalgebra generated by f is the same of the Lie algebra generated by the
k
derivations g =zh+1d of K[z].
h dz
More generally, if θ: C → D is a morphism of graded coalgebras, a morphism of graded vector
spaces d∈Homn(C,D) is called a coderivation of degree n (with respect to θ) if
∆ d=(d⊗θ+θ⊗d)∆ .
D C
In the above definition we have adopted the Koszul sign convention: i.e. if x,y ∈ C, f,g ∈
Hom∗(C,D), h,k ∈ Hom∗(B,C) are homogeneous then (f ⊗g)(x⊗y) = (−1)gxf(x)⊗g(y) and
(f ⊗g)(h⊗k)=(−1)ghfh⊗gk.
Thecoderivationsofdegreenwithrespectacoalgebramorphismθ: C →D formavectorspace
denoted Codern(C,D;θ). For simplicity of notation we denote Codern(C,C)=Codern(C,C;Id).
θ ρ
Lemma 1.14. Let C−→D−→E be morphisms of graded coalgebras. The compositions with θ and
ρ induce linear maps
ρ : Codern(C,D;θ)→Codern(C,E;ρθ), f (cid:55)→ρf;
∗
θ∗: Codern(D,E;ρ)→Codern(C,E;ρθ), f (cid:55)→fθ.
Proof. Immediate consequence of the equalities
∆ ρ=(ρ⊗ρ)∆ , ∆ θ =(θ⊗θ)∆ .
E D D C
(cid:3)
θ
Lemma1.15. LetC−→Dbemorphismsofgradedcoalgebrasandletd: C →Dbeaθ-coderivation.
Then:
(1) For every n
n
(cid:88)
∆n ◦d=( θ⊗i⊗d⊗θ⊗n−i)◦∆n.
D C
i=0
(2) If p: D →V is a system of cogenerators, then d is uniquely determined by its composition
pd: C →V.
Proof. The first item is a straightforward induction on n, using the equalities ∆n =Id⊗∆n−1 and
θ⊗i∆i−1 =∆i−1θ.
C D
For item 2, we need to prove that pd = 0 implies d = 0. Assume that there exists c ∈ C such
that dc(cid:54)=0, then there exists n such that p⊗n+1∆ndc(cid:54)=0. On the other hand
D
n
(cid:88)
p⊗n+1∆ndc=( (pθ)⊗i⊗pd⊗(pθ)⊗n−i)◦∆nc=0.
D C
i=0
(cid:3)
Exercise 1.16. A counity of a graded coalgebra is a morphism of graded vector spaces (cid:15): C →K
such that ((cid:15)⊗Id )∆ = (Id ⊗(cid:15))∆ = Id . Prove that if a counity exists, then it is unique (Hint:
C C C
((cid:15)⊗(cid:15)(cid:48))∆=?).
Exercise 1.17. Let (C,∆) be a graded coalgebra. A graded subspace I ⊂C is called a coideal if
∆(I)⊂C⊗I+I⊗C. Prove that a subspace is a coideal if and only if is the kernel of a morphism
of coalgebras.
A VOYAGE ROUND COALGEBRAS 5
Exercise 1.18. Let C be a graded coalgebra and d ∈ Coder1(C,C) a codifferential of degree 1.
Prove that the triple (L,δ,[,]), where:
L=⊕n∈ZCodern(C,C), [f,g]=fg−(−1)gfgf, δ(f)=[d,f]
is a differential graded Lie algebra.
2. Connected coalgebras
Definition 2.1. A graded coalgebra (C,∆) is called nilpotent if ∆n = 0 for n >> 0. It is called
locally nilpotent if it is the direct limit of nilpotent graded coalgebras or equivalently if C =
∪ ker∆n.
n
Example 2.2. The vector space
K[t]={p(t)∈K[t]|p(0)=0}= (cid:76) Ktn
n>0
with the coproduct
n−1
(cid:88)
∆: K[t]→K[t]⊗K[t], ∆(tn)= ti⊗tn−i,
i=1
is a locally nilpotent coalgebra. The projection K[t] → K[t], p(t) → p(t)−p(0), is a morphism of
coalgebras.
Example 2.3. Let A = ⊕A be a finite dimensional graded associative commutative K-algebra
i
and let C = A∨ = Hom∗(A,K) be its graded dual. Since A and C are finite dimensional, the
pairing (cid:104)c1⊗c2,a1⊗a2(cid:105)=(−1)a1c2(cid:104)c1,a1(cid:105)(cid:104)c2,a2(cid:105) gives a natural isomorphism C⊗C =(A⊗A)∨
commuting with the twisting maps T; we may define ∆ as the transpose of the multiplication map
µ: A⊗A→A. Then (C,∆) is a coassociative cocommutative coalgebra. Note that C is nilpotent
if and only if A is nilpotent.
Exercise 2.4. Let (C,∆) be a graded coalgebra. Prove that for every a,b≥0
a+1
∆a(ker∆a+b)⊂ (cid:78)(ker∆b).
(Hint: prove first that ∆a(ker∆a+b)⊂ker∆b⊗C⊗a.)
Exercise 2.5. Let (C,∆) be a locally nilpotent graded coalgebra. Prove that every projection
p: C →ker∆ is a system of cogenerators.
Definition 2.6 ([8, p. 282]). A graded coalgebra (C,∆) is called connected if there is an element
1∈C such that ∆(1)=1⊗1 (in particular deg(1)=0) and C =∪+∞F C, where F C is defined
r=0 r r
recursively by the formulas
F C =K1, F C ={x∈C |∆(x)−1⊗x−x⊗1∈F C⊗F C}.
0 r+1 r r
Example 2.7. Every locally nilpotent coalgebra is connected (with 1 = 0, see Exercise 2.4). If
f: C →D is a surjective morphism of coalgebras and C is connected, then also D is connected.
Lemma 2.8. Let C be a connected coalgebra and e∈C such that ∆(e)=e⊗e. Then either e=0
or e=1. In particular the idempotent 1 as in Definition 2.6 is determined by C.
Proof. Let r be the minimum integer such that e ∈ F C. If r = 0 then e = t1 for some t ∈ K; if
r
1(cid:54)=0 then t2 =t and t=0,1. If r >0 we have
(e−1)⊗(e−1)=∆(e)−1⊗e−e⊗1+1⊗1∈F C⊗F C
r−1 r−1
and then e−1∈F C which is a contradiction. (cid:3)
r−1
The reduction of a connected coalgebra C is defined as its quotient C = C/K1; it is a locally
nilpotent coalgebra.
6 MARCOMANETTI
3. The reduced tensor coalgebra
Given a graded vector space V, we denote T(V) = (cid:76) (cid:78)nV. When considered as a subset
n>0
of T(V) it becomes an ideal of the tensor algebra generated by V. The reduced tensor coalgebra
generated by V is the graded vector space T(V) endowed with the coproduct
n−1
(cid:88)
a: T(V)→T(V)⊗T(V), a(v ⊗···⊗v )= (v ⊗···⊗v )⊗(v ⊗···⊗v ).
1 n 1 r r+1 n
r=1
We can also write
+∞ n−1
(cid:88) (cid:88)
a= a ,
a,n−a
n=2 a=1
where a : (cid:78)nV →(cid:78)aV ⊗(cid:78)n−aV is the inverse of the multiplication map.
a,n−a
Thecoalgebra(T(V),a)iscoassociative,itislocallynilpotentandtheprojectionp1: T(V)→V
is a system of cogenerators: in fact, for every s>0,
(cid:88)
as−1(v ⊗···⊗v )= (v ⊗···⊗v )⊗···⊗(v ⊗···⊗v )
1 n 1 i1 is−1+1 n
1≤i1<i2<···<is=n
and then
s−1
keras−1 = (cid:76) V⊗i, (⊗sp1)as−1 =ps: T(V)→V⊗s.
i=1
Exercise3.1. Letµ: (cid:78)sT(V)→T(V)bethemultiplicationmap.Provethatforeveryv ,...,v ∈
1 n
V
(cid:18) (cid:19)
n−1
µas−1(v ⊗···⊗v )= v ⊗···⊗v .
1 n s−1 1 n
Foreverymorphismofgradedvectorspacesf: V →W theinducedmorphismofgradedalgebras
T(f): T(V)→T(W), T(f)(v ⊗···⊗v )=f(v )⊗···⊗f(v )
1 n 1 n
is also a morphism of graded coalgebras.
If (C,∆) is a locally nilpotent graded coalgebra then, for every c ∈ C, there exists n > 0 such
that ∆n(c)=0 and then it is defined a morphism of graded vector spaces
∞
1 (cid:88)
= ∆n: C →T(C).
1−∆
n=0
Proposition 3.2. Let (C,∆) be a locally nilpotent graded coalgebra, then:
1
(1) The map =(cid:80) ∆n: C →T(C) is a morphism of graded coalgebras.
1−∆ n≥0
(2) For every graded vector space V and every morphism of graded coalgebras φ: C → T(V),
there exists a unique morphism of graded vector spaces f: C →V such that φ factors as
∞
1 (cid:88)
φ=T(f) = (⊗nf)∆n−1: C →T(C)→T(V).
1−∆
n=1
Proof. [1] We have
   
n
(cid:88) (cid:88) (cid:88)(cid:88)
 ∆n⊗ ∆n∆= (∆a⊗∆n−a)∆
n≥0 n≥0 n≥0a=0
 
n
(cid:88)(cid:88) (cid:88)
= aa+1,n+1−a∆n+1 =a ∆n
n≥0a=0 n≥0
where in the last equality we have used the relation a∆0 =0.
[2] The unicity of f is clear, since by the formula φ = T(f)((cid:80) ∆n) it follows that f = p1φ.
n≥0
Toprovetheexistenceofthefactorization,takeanymorphismofgradedcoalgebrasφ: C →T(V),
denote by f = p1φ and by ψ: C → T(V) the coalgebra morphism ψ = T(f)(1−∆)−1. Since
p1ψ =p1φ and p1 is a system of cogenerators we have φ=ψ. (cid:3)
A VOYAGE ROUND COALGEBRAS 7
It is useful to restate part of the Proposition 3.2 in the following form
Corollary 3.3. Let V be a fixed graded vector space; for every locally nilpotent graded coalgebra C
the composition with the projection p1: T(V)→V induces a bijection
∼
Hom (C,T(V))−→Hom (C,V).
GC G
In other words, every morphism of graded vector spaces C →V has a unique lifting to a morphism
of graded coalgebras C →T(V).
When C is a reduced tensor coalgebra, Proposition 3.2 takes the following more explicit form
Corollary 3.4. Let U,V be graded vector spaces. the projection. Given f: T(U) → V, the linear
map F: T(U)→T(V):
n
(cid:88) (cid:88)
F(v ⊗···⊗v )= f(v ⊗···⊗v )⊗···⊗f(v ⊗···⊗v ),
1 n 1 i1 is−1+1 is
s=1 1≤i1<i2<···<is=n
is the morphism of graded coalgebras lifting f.
Example 3.5. Let A be an associative graded algebra. Consider the projection p: T(A)→A, the
multiplication map µ: T(A)→A and its conjugate
µ∗ =−µT(−1), µ∗(a ⊗···⊗a )=(−1)n−1µ(a ⊗···⊗a )=(−1)n−1a a ···a .
1 n 1 n 1 2 n
ThetwocoalgebramorphismsT(A)→T(A)inducedbyµandµ∗ areisomorphisms,theoneinverse
of the other. In fact, the coalgebra morphism F: T(A)→T(A)
n
(cid:88) (cid:88)
F(a ⊗···⊗a )= (a a ···a )⊗···⊗(a ···a )
1 n 1 2 i1 is−1+1 is
s=1 1≤i1<i2<···<is=n
is induced by µ (i.e. pF =µ), µ∗F(a)=a for every a∈A and for every n≥2
n
(cid:88) (cid:88)
µ∗F(a ⊗···⊗a )= (−1)s−1 a a ···a =
1 n 1 2 n
s=1 1≤i1<i2<···<is=n
n (cid:18) (cid:19) (cid:32)n−1 (cid:18) (cid:19)(cid:33)
(cid:88) n−1 (cid:88) n−1
= (−1)s−1 a a ···a = (−1)s a a ···a =0.
s−1 1 2 n s 1 2 n
s=1 s=0
This implies that µ∗F = p and therefore, if F∗: T(A) → T(A) is induced by µ∗ then pF∗F =
µ∗F =p and by Corollary 3.3 F∗F is the identity.
Proposition 3.6. Let (C,∆) be a locally nilpotent graded coalgebra, V a graded vector space and
∞
(cid:88)
θ = (⊗nf)∆n−1: C →T(V)
n=1
themorphismofcoalgebrasinducedbypθ =f ∈Hom0(C,V).Foreverynandeveryq ∈Homk(C,V),
the linear map
∞ n
(cid:88) (cid:88)
Q= ( (f⊗i⊗q⊗f⊗n−i)∆n: C →T(V)
n=0 i=0
is the θ-coderivation induced by pQ=q. In particular the map
Coderk(C,T(V);θ)→Homk(C,V), Q(cid:55)→pQ,
is bijective.
Proof. The map Q is the composition of the coalgebra morphism (cid:80)∆n: C →T(C) and the map
(cid:88)
R: T(C)→T(V), R= f⊗i⊗q⊗f⊗j.
i,j≥0
It is therefore sufficient to prove that R is a T(f)-coderivation, i.e. that satisfies the coLeibniz
rule
(R⊗T(f)+T(f)⊗R)a=aR.
8 MARCOMANETTI
Denoting R =(cid:80) f⊗i⊗q⊗f⊗j we have, for every a,n
n i+j=n−1
a R =(R ⊗f⊗n−a+f⊗a⊗R )a .
a,n−a n a n−a a,n−a
Taking the sum over a,n−a we get the proof. (cid:3)
Corollary 3.7. Let V be a graded vector space. Every q ∈ Homk(T(V),V) induce a coderivation
Q∈Coderk(T(V),T(V)) given by the explicit formula
Q(a ⊗···⊗a )=
1 n
(cid:88)
= (−1)k(a1+···+ai)a ⊗···⊗a ⊗q(a ⊗···⊗a )⊗···⊗a .
1 i i+1 i+l n
i,l
Proof. Apply Proposition 3.6 with the map f: T(V) → V equal to the projection (and then
θ =Id). (cid:3)
Exercise 3.8. Let p: T(V) → T(V) be the projection with kernel K = (cid:78)0V and φ: T(V) →
T(V)⊗T(V)theuniquehomomorphismofgradedalgebrassuchthatφ(v)=v⊗1+1⊗v forevery
v ∈V. Prove that pφ=ap.
Exercise 3.9. Let A be an associative graded algebra over the field K, for every local homomor-
phism of K-algebras γ: K[[x]] → K[[x]], γ(x) = (cid:80)γ xn, we can associate a coalgebra morphism
n
F : T(A)→T(A) induced by the linear map
γ
f : T(A)→A, f(a ⊗···⊗a )=γ a ···a .
γ 1 n n 1 n
Prove the composition formula F =F F . (Hint: Example 1.6.)
γδ δ γ
Exercise 3.10. A graded coalgebra morphism F: T(U) → T(V) is surjective (resp.: injective,
bijective) if and only if F1: U →V is surjective (resp.: injective, bijective). (Hint: F preserves the
1
filtrations of kernels of iterated coproducts.)
4. Rooted trees
Definition 4.1. An unreduced1 rooted forest is the data of a finite set of vertices V and a flow
map f: V →V such that:
(cid:92)
Fix(f)= fn(V −Fix(f)),
n>0
where Fix(f)={v ∈V |f(v)=v} is the subset of fixed points of f.
The vertices of an unreduced rooted forest (V,f) are divided into three disjoint classes:
• V ={root vertices}=Fix(f).
r
• V ={tail vertices}=V −f(V).
t
• V ={internal vertices}=f(V)−{root vertices}.
i
Every unreduced rooted forest (V,f) can be described by a directed graph with set of vertices
V and oriented edges v→f(v) for every v (cid:54)∈ {root vertices}. In our pictures internal vertices will
be denoted by a black dot, while tail and root vertices will be denoted by a circle. As an example,
the pair (V,f), where V ={1,2,3,4} and f(i)=min(4,i+1) is an almost rooted forest described
by the oriented graph
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
◦ •• •• ◦
tail root
Note that the map f: V ∪V → V ∪V is surjective and then the number of tail vertices is
t i i r
always greater than or equal to the number of root vertices.
The set of edges {(v,f(v))|v (cid:54)∈{roots}} is divided into types. An edge (v,f(v)) is called a root
edge if f(v) is a root vertex; it is called a tail edge if v is a tail vertex and it is called an internal
edge ifbothv,f(v)arebothinternalvertices.Noticethatanedgemaybetailandrootatthesame
time.
1Hereunreducedmeansnotnecessarilyreduced.
A VOYAGE ROUND COALGEBRAS 9
The arity (also called valence in literature) |v| of a vertex v is the number of incoming edges;
equivalently
|v|=|{w (cid:54)=v |f(w)=v}|.
Arooted forest isanalmostrootedforestsuchthateveryroothasarity1andeveryinternalvertex
has arity ≥2. A rooted tree is a rooted forest with exactly one root.
Every rooted forest is then a disjoint union of rooted trees; the following picture represents a
rooted tree with 5 tail vertices and 4 internal vertices.
◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)◦•(cid:35)(cid:35)(cid:59)(cid:59)(cid:119)(cid:71)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)•(cid:59)(cid:59)(cid:71)(cid:35)(cid:35) (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:35)(cid:35)
◦ ◦(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)(cid:59)(cid:59)••◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)•(cid:59)(cid:59)(cid:35)(cid:35)•• ◦(cid:47)(cid:47) root
◦
Anautomorphismofarootedforest(V,f)isabijectivemapφ: V →V suchthatfφ=φf.The
group of automorphisms will be denoted by Aut(V,f).
Definition 4.2. Let (V,f) be a rooted forest. An orientation of (V,f) is a total ordering ≤ on
the set V of tail vertices such that if v ≤u≤w and fk(v)=fh(w), for some h,k ≥0, then there
t
exists l ≥ 0 such that fk(v) = fl(u) = fh(w). It is often convenient to describe an orientation ≤
by the order-preserving bijection ν: {1,...,n}→V , where |V |=n. Therefore, an oriented rooted
t t
forest is a triple (V,f,ν) where ν is an orientation of (V,f).
For instance, there are (up to isomorphism) exactly three oriented rooted trees with 3 tails:
12◦◦(cid:71)(cid:119)(cid:119)(cid:119)(cid:71)(cid:119)(cid:35)(cid:35)•(cid:59)(cid:59)◦•(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)••(cid:35)(cid:35)•(cid:59)(cid:59) ◦(cid:47)(cid:47) 32◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:119)◦•(cid:35)(cid:35)•(cid:59)(cid:59)(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)•(cid:35)(cid:35)•(cid:59)(cid:59)• ◦(cid:47)(cid:47) 12◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)•(cid:47)(cid:47)••(cid:59)(cid:59)•(cid:35)(cid:35) ◦(cid:47)(cid:47)
3◦ 1◦ 3◦
Lemma 4.3. Let V be a rooted tree. Then the number of isomorphism classes of orientations on
V is equal to
1 (cid:89)
|v|!
|Aut(V)|
v∈Vi
Proof. The group of automorphisms of V acts freely on the set of orientations. We note that every
orientation is uniquely determined by:
(1) A total ordering of root edges.
(2) For every internal vertex, a total ordering of incoming edges.
Therefore the product (cid:81) |v|! is equal to the number of orientations on the tree V. (cid:3)
v∈Vi
Denote by F(n,m) the set of isomorphism classes of oriented rooted forests with n tails and m
roots. Notice that F(n,m)=∅ for m>n and F(n,n) contains only one element, denoted by I .
n
There are defined naturally two binary operations:
◦: F(l,m)×F(n,l)→F(n,m) composition
⊗: F(n,m)×F(a,b)→F(n+a,m+b) tensor product
ThetensorproductV ⊗W isthedisjointunionofV andW withtheorientation{1,2,...}→W
t
shifted by the number of tail vertices of V. For instance I ⊗I =I and
a b a+b
◦(cid:79)(cid:79) ◦(cid:79)(cid:79) ◦(cid:79)(cid:79) ◦(cid:79)(cid:79)
(cid:10)(cid:10)(cid:10)(cid:10)•••(cid:68)(cid:68)(cid:52)(cid:90)(cid:90)(cid:52)(cid:52)(cid:52) ⊗ (cid:10)(cid:10)(cid:10)(cid:10)•••(cid:68)(cid:68)(cid:90)(cid:90)(cid:52)(cid:52)(cid:52)(cid:52) = (cid:10)(cid:10)(cid:10)(cid:10)•••(cid:68)(cid:68)(cid:90)(cid:90)(cid:52)(cid:52)(cid:52)(cid:52) (cid:10)(cid:10)(cid:10)(cid:10)•••(cid:68)(cid:68)(cid:90)(cid:90)(cid:52)(cid:52)(cid:52)(cid:52)
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
1 2 1 2 1 2 3 4
Given (V,f)∈F(n,l) and (W,g)∈F(l,m) we define W ◦V in the following way: first we take
the unique bijection η: V →W such that for i>>0 the map
r t
ηfi
V −−→W
t t
10 MARCOMANETTI
is nondecreasing. Then we use η to annihilate the root vertices of V with the tail vertices of W.
For instance, for every V ∈F(n,m) we have V =I ◦V =V ◦I and
m n
◦(cid:10)(cid:10)(cid:10)(cid:10)•••◦(cid:68)(cid:68)(cid:79)(cid:79)(cid:52)(cid:90)(cid:90)(cid:52)(cid:52)(cid:52)◦ ◦ 12◦◦(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)•(cid:47)(cid:47)••(cid:35)(cid:35) ◦(cid:47)(cid:47)(cid:47)(cid:47) = 12◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:119)◦•(cid:35)(cid:35)•(cid:59)(cid:59)(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)•(cid:35)(cid:35)•(cid:59)(cid:59)• ◦(cid:47)(cid:47)
1 2 3◦ ◦ 3◦
The operations ◦ and ⊗ are associative and satisfy the interchange law [6]: this means that
(V ⊗W)◦(A⊗B)=(V ◦A)⊗(W ◦B)
holdswheneverthecompositesV ◦AandW◦B aredefined.ByconventionwesetI =∅∈F(0,0)
0
and then I ⊗V =V ⊗I =V for every V.
0 0
Exercise 4.4. Given V ∈F(n,m) denote by
w(V)=max{a|∃W ∈F(n−a,m−a) such that V =I ⊗W}.
a
WeshallsaythatacompositionV ◦V ◦···◦V ismonotone ifw(V )≤w(V )≤···≤w(V ).Prove
1 2 r 1 2 r
that every oriented rooted forest V ∈F(n,m) can be written uniquely as a monotone composition
of oriented rooted forests with one internal vertex.
5. Automorphisms of T(V) and inversion formula.
For every graded vector space V we can define binary operations
◦: Hom∗(V⊗l,V⊗m)×Hom∗(V⊗n,V⊗l)→Hom∗(V⊗n,V⊗m) (f,g)(cid:55)→f ◦g,
⊗: Hom∗(V⊗n,V⊗m)×Hom∗(V⊗a,V⊗b)→Hom∗(V⊗n+a,V⊗m+b) (f,g)(cid:55)→f ⊗g.
By a representation of F =∪ F(n,m) we shall mean a map
n,m
(cid:91)
Z: F → Hom∗(V⊗n,V⊗m)
n,m
such that ZIn = IdV⊗n and commutes with the operations ◦ and ⊗. Every representation Z is
determined by its value on the irreducible trees T . Conversely, for every sequence of maps f ∈
n n
Hom∗(V⊗n,V), n≥2, there exists an unique representation
(cid:91)
Z(f ): F → Hom∗(V⊗n,V⊗m)
i
n,m
such that
ZTn(fi)=fn.
For instance, the oriented rooted tree
1◦(cid:71)(cid:71)(cid:35)(cid:35)
Γ= 2◦(cid:119)(cid:119)(cid:119)(cid:119)◦••(cid:59)(cid:59)(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)•(cid:35)(cid:35)•(cid:59)(cid:59)• ◦(cid:47)(cid:47)
3◦
gives
Z (f )(v ⊗v ⊗v )=f (f (v ⊗v )⊗v ),
Γ i 1 2 3 2 2 1 2 3
while the oriented rooted forest
3◦(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:35)(cid:35)(cid:47)(cid:47) (cid:47)(cid:47)
Γ= 2◦ ••• ◦
(cid:47)(cid:47)
1◦ ◦
gives
Z (f )(v ⊗v ⊗v )=(−1)deg(v1)deg(f2)v ⊗f (v ⊗v ).
Γ i 1 2 3 1 2 2 3
Definition 5.1. For every n,m let S(n,m) ⊂ F(n,m) be the subset of (isomorphism classes of)
(cid:83)
oriented rooted forests without internal edges and denote S = S(n,m).
n,m
Equivalently Γ∈S if and only if Γ is the tensor product of irreducible oriented rooted trees.